Download Chapter 6 Long-run aspects of fiscal policy and

Chapter 6
Long-run aspects of …scal policy
and public debt
We consider an economy with a government providing public goods and services.
It …nances its spending by taxation and borrowing. The term …scal policy refers to
policy that involves decisions about the government’s spending and the …nancing
of this spending, be it by taxes or debt issue. The government’s choice concerning
the level and composition of its spending and how to …nance it, may aim at:
1 a¤ecting resource allocation (provide public goods that would otherwise not
be supplied in a su¢ cient amount, correct externalities and other markets
failures, prevent monopoly ine¢ ciencies, provide social insurance);
2 a¤ecting income distribution, be it a) within generations or b) between
generations;
3 contribute to macroeconomic stabilization (dampening of business cycle
‡uctuations through aggregate demand policies).
The design of …scal policy with regard to the aims 1 and 2 at a disaggregate
level is a major theme within the …eld of public economics. Macroeconomics deals
with aim 3 and the big-picture aspects of 1 and 2, like overall policy to enhance
economic growth.
In this chapter we address the issue of …scal sustainability and long-run implications of debt …nance. This relates to one of the conditions that constrain
public …nancing instruments. To see the issue of …scal sustainability in a broader
context, Section 6.1 provides an overview of conditions and factors that constrain
public …nancing instruments. Section 6.2 introduces the basics of government
budgeting and Section 6.3 de…nes the concepts of government solvency and …scal
sustainability. In Section 6.4 the analytics of debt dynamics is presented. As an
201
202
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
example the Stability and Growth Pact of the EMU (the Economic and Monetary Union of the European Union) is discussed. Section 6.5 looks more closely
at the link between government solvency and the government’s No-Ponzi-Game
condition and intertemporal budget constraint. This is applied in Section 6.6 to
a study of the Ricardian equivalence proposition; applying the Diamond OLG
framework we address the question: Is Ricardian equivalence likely to be a good
approximation to reality? If not, why?
6.1
An overview of government …nancing issues
Before entering the more specialized sections, it is useful to have a general idea
about circumstances that constrain public …nancing instruments. These circumstances include:
(i) …nancing by debt issue is constrained by the need to remain solvent and
avoid catastrophic debt dynamics;
(ii) …nancing by taxes is limited by problems arising from:
(a) distortionary supply-side e¤ects of many kinds of taxes;
(b) tax evasion (cf. the rise of the shadow economy, tax havens used by
multinationals, etc.).
(iii) time lags (such as recognition lag, decision lag, implementation lag, and
e¤ect lag);
(iv) credibility problems due to time-inconsistency;
(v) limitations imposed by political processes, bureaucratic self-interest, lobbying, and rent seeking.
Point (i) is discussed in detail in the sections below. The remaining points,
(ii) (v) are not addressed speci…cally in this chapter. They should always be
kept in mind, however, when discussing …scal policy. Hence a few remarks here.
First, the points (ii.a) and (ii.b) give rise to what is known as the La¤er curve
(after the American economist Arthur La¤er, 1940-). The La¤er curve refers to a
hump-shaped relationship between the income tax rate and the tax revenue. For
simplicity, suppose the tax revenue equals taxable income times a given average
tax rate. A 0% tax rate and most likely also a 100% tax rate generate no tax
revenue. As the tax rate increases from a low initial level, a rising tax revenue is
obtained. But after a certain point some people may begin to work less (in the
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.2. The government budget
203
legal economy), stop reporting all their income, and stop investing. While La¤er
was seemingly wrong about where USA was on the curve (see, e.g., Fullerton
1982), surely there is a point above which tax revenue depends negatively on the
tax rate. This notwithstanding, in practice there is no such thing as the La¤er
curve. This is because a lot of contingencies are involved. Income taxes are
typically progressive1 so that marginal tax rates di¤er from average tax rates; it
matters how the tax revenue is spent by the government, etc.
That time lags, point (iii), are a constraining factor is especially important
for macroeconomic stabilization policy aiming at dampening business cycle ‡uctuations. If these potential lags are ignored, there is a risk that the government
intervention comes too late and ends up amplifying the ‡uctuations instead of
dampening them. In particular the monetarists, lead by the American economist
Milton Friedman (1912-2006), warned against this risk.
Point (iv) hints at the fact that when outcomes depend on forward-looking
expectations in the private sector, governments sometimes face what is known
as the time-inconsistency problem. Time-inconsistency refers to the possible
temptation of the government to deviate from its previously announced course
of action once the private sector has acted. An example: With the purpose
of stimulating private saving, the government announces that it will not tax
…nancial wealth. Nevertheless, when …nancial wealth has reached a certain level,
it constitutes a tempting base for taxation and so a tax on wealth might be
levied. To the extent the private sector anticipates this, the attempt to a¤ect
private saving in the …rst place fails. We return to this kind of problems in other
chapters.
Finally, political processes, bureaucratic self-interest, lobbying, and rent seek2
ing may interfere with …scal policy. This is a theme in the branch of economics
called political economy and is outside the scope of this chapter.
Now to the speci…cs of government budget accounting and debt …nancing.
6.2
The government budget
We generally perceive the public sector as consisting of the government (at national as well as local level) and a central bank. In economics the term “government”is used in a broad sense, encompassing both legislation and administration
concerning spending on public consumption and investment, levying taxes, paying transfers and subsidies, and paying interest on government debt. Within
1
A tax code in which average tax rates rise with income is called progressive.
Rent seeking refers to attempts to gain by increasing one’s share of existing wealth, instead
of trying to produce wealth.
2
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
204
certain limits the government has usually delegated the management of the nation’s currency to the central bank, also called the monetary authority. Until
further notice, our accounting treats “government budgeting” as covering the
public sector as a whole, that is, the consolidated government and central bank.
Government bonds held by the central bank are thus excluded from what we
call “government debt”. So the terms government debt and public debt are used
synonymously.
The basics of government budget accounting cannot be described without
including money, nominal prices, and in‡ation. Elementary aspects of money
and in‡ation will therefore be included in this section. We shall not, however,
consider money and in‡ation in any systematic way until later chapters.
Table 6.1 lists key variables of government budgeting, assuming a closed economy.
Symbol
Yt
Ctg
Itg
Gt
Xt
T~t
Tt T~t Xt
Mt
Pt
Dt
Bt PDt t1
bt BYtt
it
xt
t
1 + rt
Pt
Pt 1
Pt 1 (1+it )
Pt
Table 6.1. List of main variable symbols
Meaning
real GDP = real GNP (closed economy)
public consumption
public …xed capital investment
Ctg + Itg real public spending on goods and services
real transfer payments
real gross tax revenue
real net tax revenue
the monetary base (currency and bank reserves in the central bank)
price level (in money) for goods and services (the GDP de‡ator)
nominal net public debt
real net public debt
government debt-to-income ratio
nominal short-term interest rate
= xt xt 1 (where x is some arbitrary variable)
Pt Pt 1
in‡ation rate
Pt 1
1+it
1+ t
real short-term interest rate
Note that Yt ; Gt ; and Tt are quantities de…ned per period, or more generally,
per time unit, and are thus ‡ow variables. On the other hand, Mt ; Dt ; and Bt
are stock variables, that is, quantities de…ned at a given point in time, here at
the beginning of period t. We measure Dt and Bt net of …nancial claims held
by the government. Almost all countries have positive government net debt, but
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.2. The government budget
205
in principle Dt < 0 is possible.3 The monetary base, Mt ; is currency plus fully
liquid deposits in the central bank held by the private sector at the beginning of
period t; Mt is by de…nition nonnegative.
Ignoring, at least to begin with, uncertainty and risk of default, the nominal
interest rate on government bonds must be the same as that on other interestbearing assets in the economy. For ease of exposition we imagine that all government bonds are one-period bonds. That is, each government bond promises a
payout equal to one unit of account at the end of the period and then the bond
expires. Given the interest rate, it ; the market value of a bond at the start of
period t is vt = 1=(1 + it ): If the number of outstanding bonds (the quantity of
bonds) in period t is qt ; the government debt has face value (value at maturity)
equal to qt . The market value at the start of period t of this quantity of bonds
will be Dt = qt =(1 + it ). The nominal expenditure to be made at the end of the
period to redeem the outstanding debt can then be written
qt = Dt (1 + it ):
(6.1)
This is the usual way of writing the expenditure to be made, namely as if the
government debt were like a given bank loan of size Dt with a variable rate of
interest. We should not forget, however, that given the quantity, qt ; of the bonds,
the value, Dt , of the government debt at the issue date depends negatively on it :
Anyway, the total nominal government expenditure in period t can be written
Pt (Gt + Xt ) + Dt (1 + it ):
It is common to refer to this expression as expenditure “in period t”. Yet, in a
discrete time model (with a period length of a year or a quarter corresponding
to typical macroeconomic data) one has to imagine that the payment for goods
and services delivered in the period occurs either at the beginning or the end of
the period. We follow the latter interpretation and so the nominal price level Pt
for period-t goods and services refers to payment occurring at the end of period
t: As an implication, the real value, Bt ; of government debt at the beginning of
period t (= end of period t 1) is Dt =Pt 1 : This may look a little awkward but
is nevertheless meaningful. Indeed, Dt is a stock of liabilities at the beginning
of period t while Pt 1 is a price referring to a ‡ow paid for at the end of period
t 1 which is essentially the same point in time as the beginning of period t:
Anyway, whatever timing convention is chosen, some kind of awkwardness will
always arise in discrete time analysis. This is because the discrete time approach
3
If Dt < 0, the government has positive net …nancial claims on the private sector and earns
interest on these claims
which is then an additional source of government revenue besides
taxation.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
206
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
arti…cially treats the continuous ‡ow of time as a sequence of discrete points in
time.4
The government expenditure is …nanced by a combination of taxes, bonds
issue, and change in the monetary base:
Pt T~t + Dt+1 +
Mt+1 = Pt (Gt + Xt ) + Dt (1 + it ):
(6.2)
By rearranging we have
Dt+1 +
Mt+1 = Pt (Gt + Xt
T~t ) + it Dt :
(6.3)
In standard government budget accounting the nominal government budget
de…cit, GBD; is de…ned as the excess of total government spending over government revenue, P T~. That is, according to this de…nition the right-hand side of
(6.3) is the nominal budget de…cit in period t; GBDt : The …rst term on the righthand side, Pt (Gt + Xt T~t ); is named the primary budget de…cit (non-interest
spending less taxes). The second term, it Dt ; is called the debt service. Similarly,
Pt (T~t Xt Gt ) is called the primary budget surplus. Note that negative values of
“de…cits”and “surpluses”represent positive values of “surpluses”and “de…cits”,
respectively.
We immediately see that this budget accounting deviates from “regular”budgeting principles. Private companies, for instance, typically have separate capital
and operating budgets. In contrast, the budget de…cit de…ned above treats that
part of G which represents government net investment parallel to government
consumption. Government net investment is attributed as a cost in a single
year’s account; according to “normal” budgeting principles it is only the depreciation on the public capital that should …gure as a cost. Likewise the above
budget accounting does not consider that a part of D (or perhaps more than D)
may be backed by the value of public physical capital. And if the government
sells a physical asset to the private sector, it will appear as a reduction of the
government budget de…cit while in reality it is merely a conversion of an asset
from a physical form to a …nancial form. That is, the cost aspects, belonging
to the operating budget, and the asset aspects, belonging to the capital budget,
of government net investment are not properly dealt with in the standard public
accounting.5
4
In a theoretical model this kind of problems is avoided when government budgeting is
formulated in continuous time, cf. Chapter 13.
5
Moreover, some countries, including Denmark, have large implicit government assets due
to deferred taxes on the part of personal income invested in pension funds. If the government
then decides to reverse the deferred taxation (as the Danish government did 2012 and 2014
to comply better with the 3%-de…cit rule of the Stability and Growth Pact of the EMU), the
o¢ cial budget de…cit is reduced, but essentially it is a matter of replacing one government asset
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.2. The government budget
207
In our elementary treatment below we will nevertheless stick to the traditional
vocabulary. Where this might create logical problems (as it almost inevitably will
do), it helps to imagine that:
(a) all of G is public consumption, i.e., Gt = Ctg for all t;
(b) there is no public physical capital.
An additional anomaly relates to what the accounting usually includes under
“public consumption”. A sizeable part of this is in fact investment in an economic
sense: expenses on education, research, and health. In this chapter, however,
distinguishing between these categories and public consumption in a narrow sense
(administration, judicial system, police, defence) is unimportant.
Now, from (6.3) and the de…nition Tt T~t Xt (net tax revenue) follows that
real government debt at the beginning of period t + 1 is:
Bt+1
Dt
Dt+1
= Gt + Xt T~t + (1 + it )
Pt
Pt
1 + it
Mt+1
= Gt Tt +
Bt
1+ t
Pt
Mt+1
:
(1 + rt )Bt + Gt Tt
Pt
Mt+1
Pt
(6.4)
The last term, Mt+1 =Pt ; in this expression is seigniorage, i.e., public sector
revenue obtained by issuing base money.
Suppose real output grows at the constant rate gY so that Yt+1 = (1 + gY )Yt :
Then the public debt-to-income ratio can be written
bt+1
Bt+1
1 + rt
Gt Tt
=
bt +
Yt+1
1 + gY
(1 + gY )Yt
Mt+1
:
Pt (1 + gY )Yt
(6.5)
The last term here is the (growth-corrected) seigniorage-income ratio,
Mt+1
= (1 + gY )
Pt (1 + gY )Yt
1
Mt+1 Mt
:
Mt Pt Yt
If in the long run the base money growth rate, Mt+1 =Mt , and the nominal interest rate (i.e., the opportunity cost of holding money, cf. Chapter 16) are constant,
then the velocity of money and its inverse, the money-income ratio, Mt =(Pt Yt ),
are also likely to be more or less constant. So is, therefore, the seigniorage-income
by another.
In Chapter 13 we consider a government accounting framework with separate capital and
operating budgets.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
208
ratio. For more developed countries this ratio is generally a small number.6 For
some of the emerging economies that have poor institutions for collection of taxes
it is considerably higher and time-varying. And in situations of budget crisis
and hyperin‡ation it may be sizeable (see Montiel, 2003).
Generally budget de…cits are …nanced primarily by debt creation but also to
some extent by money creation, as envisioned in the above equations.7 Nonetheless, from now on in this chapter we will in the main ignore the seigniorage term
in (6.5) since it tends to be small in developed countries.
We thus proceed with the simple government accounting equation:
Bt+1
Bt = rt Bt + Gt
Tt ;
(DGBC)
where the right-hand side is the real budget de…cit. This equation is in macroeconomics often called the dynamic government budget constraint (or DGBC for
short) although it is in itself just an accounting identity conditional on M = 0.
It says that if the real budget de…cit is positive and there is essentially no …nancing by money creation, then the real public debt grows. We come closer to a
constraint when combining (DGBC) with the requirement that the government
stays solvent.
6.3
Government solvency and …scal sustainability
To be solvent means being able to meet the …nancial commitments as they fall
due. In practice this concept is closely related to the government’s No-PonziGame condition and intertemporal budget constraint (to which we return in Section 6.5), but at the theoretical level it is more fundamental.
We may view the public sector as an in…nitely-lived agent in the sense that
there is no last date where all public debt has to be repaid. Nevertheless, as we
shall see, there tends to be stringent constraints on government debt creation in
the long run.
6
In Denmark seigniorage was around 0.2 per cent of GDP during the 1990s (Kvartalsoversigt
4. kvartal 2000, Danmarks Nationalbank).
7
The existing …scal-…nancial framework for the Eurozone countries o¢ cially excludes the
European Central Bank (ECB). Yet, for the EMU area as a whole, seigniorage is, of course,
income, albeit small, for the aggregate consolidated public sector. The seigniorage is every year
divided by the national central banks of the Eurozone countries in proportion to their equity
share in the ECB. And the central banks then transfer their share to the national treasuries.
The Amsterdam Treaty of 1997 (also a part of the EMU) prohibits the national central banks
from “actively” providing monetary …nancing for public de…cits. The distributed seigniorage
exists nevertheless and implies that Mt+1 > 0 for the individual Eurozone countries. Yet,
Mt+1 is small relative to the countries’GDP.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.3. Government solvency and …scal sustainability
6.3.1
209
The critical role of the growth-corrected interest
rate
Very much depends on whether the real interest rate in the long-run is higher
than the growth rate of GDP or not. To see this, suppose the country considered
has positive government debt at time 0 and that the government levies taxes
equal to its non-interest spending:
T~t = Gt + Xt
or Tt
T~t
Xt = Gt for all t
0:
(6.6)
This means that taxes cover only the primary expenses while interest payments
(and debt repayments when necessary) are …nanced by issuing new debt. That is,
the government attempts a permanent roll-over of the debt including the interest
due for payment. This implies that the debt grows at the rate rt according to
(DGBC).
Assuming, for simplicity, that rt = r (a constant), the law of motion for the
public debt-to-income ratio is
bt+1
1 + r Bt
Bt+1
=
Yt+1
1 + gY Yt
1+r
bt ;
1 + gY
b0 > 0;
where we have maintained the assumption of a constant output growth rate, gY .
The solution to this linear di¤erence equation is
bt = b0 (
1+r t
);
1 + gY
where we consider both r and gY as exogenous. We see that the growth-corrected
1+r
interest rate, 1+g
1 r gY (for gY and r “small”) plays a key role: There
Y
are two contrasting cases to discuss:
Case I: r > gY : In this case, bt ! 1 for t ! 1: But this is not a feasible
path because, due to compound interest, the debt grows so large in the long run
that the government will be unable to …nd buyers for all the debt. Imagine for
example an economy described by the Diamond OLG model. Then the buyers of
the debt are the young who place part of their saving in government bonds. But
if the stock of these bonds grows at a higher rate than income, the saving of the
young cannot in the long run keep track with the fast-growing government debt.
In this situation the private sector will understand that bankruptcy is threatening
and nobody will buy government bonds except at a low price, which means a high
interest rate. The high interest rate only aggravates the problem. That is, the
…scal policy (6.6) breaks down. Either the government defaults on the debt or T
must be increased or G decreased (or both) until the growth rate of the debt is
no longer higher than gY :
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
210
If the debt is denominated in the country’s own currency, an alternative way
out is of course a shift to money …nancing of the budget de…cit, that is, seigniorage. When capacity utilization is high, this leads to accelerating in‡ation and
thus the real value of the debt is eroded. Bond holders will then demand a higher
nominal interest rate, thus aggravating the …nancing di¢ culties. The economic
and social chaos of hyperin‡ation threatens. And this …nancing route comes to
a dead end if seigniorage reaches the backward-bending part of the “seigniorage
La¤er curve”.8
Case II: r
gY : If r = gY ; we get bt = b0 for all t
0: Since the debt,
rising at the rate r; does not in the long run increase faster than national income,
the government has no problem …nding buyers of its newly issued debt. Thereby
the government is able to …nance its interest payments
it stays solvent. The
growing debt is passed on to ever new generations with higher income and saving
and the debt roll-over implied by (6.6) can continue forever. If r < gY ; we get
bt ! 0 for t ! 1; and the same conclusion holds a fortiori.
The government can thus pursue a permanent debt roll-over policy as implied
by (6.6) and still remain solvent if in the long run the interest rate is not higher
than the growth rate of the economy. But in the opposite case, permanent debt
roll-over is not possible and sooner or later at least part of the interest payments
must be tax …nanced.
Which of the two cases is relevant in real life? Fig. 6.1 shows for Denmark
(upper panel) and the US (lower panel) the time paths of the real short-term
interest rate and the GDP growth rate, both on an annual basis. Overall the
levels of the two are more or less the same, although on average the interest rate
is in Denmark slightly higher but in the US somewhat lower than the growth
rate.
Nevertheless, many macroeconomists believe there is good reason for paying
attention to the case r > gY , also for a country like the US. This is because we live
in a world of uncertainty, with many di¤erent interest rates, and imperfect credit
markets, aspects the above line of reasoning has not incorporated. Indeed, the
prudent debt policy needed in the certainty case r > gY can be shown to apply
for a larger range of circumstances when uncertainty is present (see Abel et al.
1989, Bohn 1995, Ball et al. 1998, Blanchard and Weil 2001). To give a ‡avor we
may say that a prudent debt policy is needed when the average r for the public
debt exceeds gY " for some “small”but positive ":9 On the other hand there is
a di¤erent feature which draws the matter somewhat in the opposite direction.
8
See Chapter 18.
This is only a “rough” indicator. Under uncertainty matters are for instance not so easy
that the r in the inequality r < gY characterizing Case II can be interpreted as the riskless
interest rate (Blanchard and Weil, 2001).
9
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.3. Government solvency and …scal sustainability
211
%
30
Real short-term DK interest rate (ex post)
Real annual DK GDP growth rate
20
10
0
−10
−20
Average interest rate (1875-2003):
2.9%
Compound annual GDP growth rate (1875-2002):2.7%
−30
1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
%
30
Real short-term US interest rate (ex post)
Real annual US GDP growth rate
20
10
0
−10
−20
Average interest rate (1875-2003):
2.4%
Compound annual GDP growth rate (1875-2002):3.4%
−30
1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Figure 6.1: Real short-term interest rate and annual growth rate of real GDP in Denmark and the US since 1875. The real short-term interest rate is calculated as the
money market rate minus the contemporaneous rate of consumer price in‡ation. Source:
Abildgren (2005) and Maddison (2003).
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
212
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
This is the possibility that a tax, 2 (0; 1); on interest income is in force so that
the net interest rate on the government debt is (1
)r rather than r:
6.3.2
Sustainable …scal policy
The concept of sustainable …scal policy is closely related to the concept of government solvency. As already noted, to be solvent means being able to meet the
…nancial commitments as they fall due. A given …scal policy is called sustainable
if by applying its spending and tax rules forever, the government stays solvent.
“Sustainable” conveys the intuitive meaning. The issue is: can the current tax
and spending rules continue forever?
To be speci…c, suppose Gt and Tt are determined by …scal policy rules represented by the functions
Gt = G(x1t ; :::; xnt ; t);
and
Tt = T (x1t ; :::; xnt ; t);
where t = 0; 1; 2; : : : ; and x1t ,..., xnt are key macroeconomic and demographic
variables (like net national income, old-age dependency ratio, rate of unemployment, extraction of natural resources, say oil in the ground, etc.). In this way
a given …scal policy is characterized by the rules G( ) and T ( ): Suppose further
that we have an economic model, M, of how the economy functions.
DEFINITION Let the current period be period 0 and let the public debt at
the beginning of period 0 be given. Then, given a forecast of the evolution
of the demographic and foreign economic environment in the future and given
the economic model M, the …scal policy (G( ); T ( )) is said to be sustainable
relative to this model if the forecast calculated on the basis of M is that the
government stays solvent under this policy. The …scal policy (G( ); T ( )) is called
unsustainable, if it is not sustainable.
This de…nition of …scal sustainability is silent about the presence of uncertainty. Without going into detail about this di¢ cult issue, suppose the model
M is stochastic and let " be a “small” positive number. Then we may say that
the …scal policy (G( ); T ( )) with 100-" percent probability is sustainable relative
to the model M if the forecast calculated on the basis of M is that with 100-"
percent probability the government stays solvent under this policy.
Governments, rating agencies, and other institutions evaluate sustainability
of …scal policy on the basis of simulations of giant macroeconometric models.
Essentially, the operational criterion for sustainability is whether the …scal policy
can be deemed compatible with boundedness of the public debt-to-income ratio.
Normally, the income measure applied here is GDP. Other measures are conceivable such as GNP, taxable income, or after-tax income. Moreover, even if a debt
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.4. Debt arithmetic
213
spiral is not underway in a given country, a high level of the debt-income ratio
may in itself be worrisome. This is because a high level of debt under certain
conditions may trigger a spiral of self-ful…lling expectations of default. We come
back to this in Section 6.4.1.
Owing to the increasing pressure on public …nances caused by reduced birth
rates, increased life expectancy, and a fast-growing demand for medical care,
many industrialized countries have for a long time been assessed to be in a situation where their …scal policy is not sustainable (Elmendorf and Mankiw 1999).
The implication is that sooner or later one or more expenditure rules and/or tax
rules (in a broad sense) will probably have to be changed.
Two major kinds of strategies have been suggested. One kind of strategy is
the pre-funding strategy. The idea is to prevent sharp future tax increases by
ensuring a …scal consolidation prior to the expected future demographic changes.
Another strategy (alternative or complementary to the former) is to attempt a
gradual increase in the labor force by letting the age limits for retirement and
pension increase along with expected lifetime
this is the indexed retirement
strategy. The …rst strategy implies that current generations bear a large part
of the adjustment cost. In the second strategy the costs are shared by current
and future generations in a way more similar to the way the bene…ts in the form
of increasing life expectancy are shared. We shall not here go into detail about
these matters, but refer the reader to a large literature about securing …scal
sustainability in the ageing society, see Literature notes.
6.4
Debt arithmetic
A key tool for evaluating …scal sustainability is debt arithmetic, i.e., the analytics of debt dynamics. The previous section described the important role of
the growth-corrected interest rate. The next subsection considers the minimum
primary budget surplus required for …scal sustainability in di¤erent situations.
6.4.1
The required primary budget surplus
From the dynamic government budget identity with
this context it can be ignored) we have:
Bt+1 = (1 + r)Bt
(Tt
Gt );
Mt+1 “small” (so that in
(DGBC)
where Tt Gt is the primary surplus in real terms. Let the spending-to-income
ratio, Gt =Yt , and the (net) tax revenue-to-income ratio, Tt =Yt ; be constants,
and ; respectively. It follows that the public debt-to-income ratio bt
Bt =Yt
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
214
(from now just denoted debt-income ratio) changes over time according to
bt+1
Bt+1
1+r
=
bt
Yt+1
1 + gY
1 + gY
(6.7)
;
where we have assumed that also gY and r are constant. There are three cases
to consider.
Case 1: r > gY : As emphasized above this case is generally considered the one
of most practical relevance. And it is in this case that latent debt instability is
present and the government has to pay attention to the danger of runaway debt
dynamics. To see this, note that the solution of the linear di¤erence equation
(6.7) is
bt = (b0
b
=
b)
1 + gY
1+r
1 + gY
1
t
+b ;
1+r
1 + gY
where
(6.8)
1
=
s
r
gY
r
gY
;
(6.9)
where s is the primary surplus as a share of GDP. Here b0 is historically given. But
the steady-state debt-income ratio, b ; depends on …scal policy. The important
feature is that the growth-corrected interest factor is in this case higher than
1 and has the exponent t. Therefore, if …scal policy is such that b < b0 ; the
debt-income ratio exhibits exponential growth. The solid curve in the topmost
panel in Fig. 6.2 shows a case where …scal policy is such that
< (r gY )b0
whereby we get b < b0 when r > gY ; so that the debt-income ratio, bt ; grows
without bound.
The American economist and Nobel Prize laureate George Akerlof (2004, p.
6) came up with this analogy:
“It takes some time after running o¤ the cli¤ before you begin to fall.
But the law of gravity works, and that fall is a certainty”.
Somewhat surprisingly, perhaps, when r > gY , there can be debt explosion in
the long run even if > , namely if 0 <
< (r gY )b0 : Debt explosion can
even arise if b0 < 0; namely if
< (r gY )b0 < 0:
The only way to avoid the snowball e¤ects of compound interest when the
growth-corrected interest rate is positive is to ensure a primary budget surplus as
a share of GDP,
; high enough such that b
b0 : So the minimum primary
surplus as a share of GDP, s^; required for …scal sustainability is the one implying
b = b0 ; i.e., by (6.9),
s^ = (r gY )b0 :
(6.10)
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.4. Debt arithmetic
215
Figure 6.2: Evolution of the debt-income ratio in the cases r > gY (the three upper
panels) and r < gY (the two lower panels), respectively.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
216
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
If by adjusting and/or , the government obtains
= s^; then b = b0
whereby bt = b0 for all t 0 according to (6.8), cf. the second from the top panel
in Fig. 6.2.
Note that s^ will be larger:
- the higher is the initial level of debt, b0 ; and,
- when b0 > 0; the higher is the growth-corrected interest rate, r
gY .
Delaying the adjustment increases the size of the needed policy action, since
the debt-income ratio, and thereby s^; will become higher in the meantime.
For …xed spending-income ratio ; the minimum tax-income ratio needed for
…scal sustainability is
^=
+ (r
gY )b0 :
The di¤erence, ^
; between this needed tax-to-income ratio and the actual one
indicates the size of the needed adjustment, were it to take place at time 0. A
more involved measure of the sustainability gap, taking the “room for manoeuvre”
into account, is (^ )=(1 ); this measure indicates the required increase in the
tax-income ratio relative to the “room of manoeuvre” perceived as the distance
from the actual tax-income ratio, ; up to the maximal tax-income ratio, 1: With
a La¤er curve in mind, in this formula it would make sense to replace the number
1 by a tax rate estimated to maximize the tax revenue in the country considered,
cf. Box 6.1.
Suppose that the debt build-up can be - and is - prevented already from time
0 by ensuring that the primary surplus as a share of income,
; at least equals
s^ so that b
b0 : The solid curve in the midmost panel in Fig. 6.2 illustrates the
resulting evolution of the debt-income ratio if b is at the level corresponding to
the hatched horizontal line while b0 is unchanged compared with the top panel.
Presumably, the government would in such a state of a¤airs relax its …scal policy
after a while in order not to accumulate large government …nancial net wealth.
Yet, the pre-funding strategy vis-a-vis the …scal challenge of population ageing
(referred to above) is in fact based on accumulating some positive public …nancial
net wealth as a bu¤er before the substantial e¤ects of population ageing set in. In
this context, the higher the growth-corrected interest rate, the shorter the time
needed to reach a given positive net wealth position.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.4. Debt arithmetic
217
Box 6.1. A sustainability gap measure.
Taking the room for manoeuvre into account, a natural indicator of the sustainability gap for a given spending-income ratio, , is (^
)=( L
);
where L is a tax-income ratio estimated to maximize the tax revenue in the
country considered. As noted in the …rst section of this chapter, however,
determining such a tax-income ratio is not easy because there are many
uncertainties and contingencies involved in the construction of a La¤er curve.
Let us nevertheless imagine there is a constant economy-wide tax-income
ratio that makes it meaningful to consider aggregate net income, Q; as
a function of 1
: Tax revenue is R( ) =
Q(1
); which we
assume is a hump-shaped function of in the interval [0; 1] : Taking logs
and di¤erentiating w.r.t. gives R0 ( )=R( ) = 1=
Q0 (1
)=Q(1
):
The tax-income ratio, L ; that maximizes R; satis…es 1= L
= Q0 (1
L )=Q(1
L ): Multiplying through by (1
L ) and ordering
gives
L
=
1+E`1
If the elasticity of income w.r.t. 1
1
Q(1
L)
:
is given as 0.4, we get
L
= 5=7
0:7:
Case 2: r = gY : In this knife-edge case there is still a danger of runaway
dynamics, but less explosive. The formula (6.8) is no longer valid. Instead the
solution of (6.7) is bt = b0 + [(
)=(1 + gY )] t = b0 [(
)=(1 + gY )] t: Here,
a non-negative primary surplus is both necessary and su¢ cient to avoid bt ! 1
for t ! 1.
Case 3: r < gY : This is the case of stable debt dynamics. The formula (6.8)
is again valid, but now implying that the debt-income ratio is non-explosive.
Indeed, bt ! b for t ! 1; whatever the level of the initial debt-income ratio
and whatever the sign of the budget surplus. Moreover, when r < gY ;
b =
r
gY
S 0 for
T 0:
(*)
So, if there is a forever positive primary surplus, the result is a negative long-run
debt, i.e., a positive government …nancial net wealth in the long run. And if there
is a forever negative primary surplus, the result is not debt explosion but just
convergence toward some positive long-run debt-income ratio. The second from
bottom panel in Fig. 6.2 illustrates this case for a situation where b0 > b and
b > 0; i.e.,
< 0; by (*). When the GDP growth rate continues to exceed
the interest rate on government debt, a large debt-income ratio can be brought
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
218
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
down quite fast, as witnessed by the evolution of both UK and US government
debt in the …rst three decades after the second world war. Indeed, if the growthcorrected interest rate remains negative, permanent debt roll-over can handle the
…nancing, and taxes need never be levied.10
Finally, the bottom panel in Fig. 6.2 shows the case where, with a large
primary de…cit (
< 0 but large in absolute value), excess of output growth
over the interest rate still implies convergence towards a constant debt-income
ratio, albeit a high one.
The level of the debt-income ratio and self-ful…lling expectations of
default
We here return to Case 1: r > gY : As the incumbent chief economist at the
IMF, Olivier Blanchard remarked in the midst of the 2010-2012 debt crisis in the
Eurozone:
“The higher the level of debt, the smaller is the distance between
solvency and default”.11
There will generally be an upper bound for the tax-income ratio deemed feasible by the government (think of the limits for the tax revenue implied by the
La¤er curve, say). Similarly, a lower bound for the spending-income ratio exists,
be it for economic or political reasons. In the present framework we therefore
let the government face the constraints
and
; where is the least
upper bound for the tax-income ratio and is the greatest lower bound for the
spending-income ratio. Then the actual primary surplus, s; can at most equal
s
:
Suppose that at …rst the situation in the considered country is as in the second
from the top panel in Fig. 6.2. That is, initially,
s=
= s^ = (r
gY )b0
s
;
(6.11)
with b0 > 0: De…ne r to be the value of r satisfying
(r
gY )b0 = s; i.e., r =
s
+ gY :
b0
(6.12)
Thereby r is the maximum level of the interest rate consistent with absence of
an explosive debt-income ratio.
10
On the other hand, we should not forget that this analysis presupposes absence of uncertainty. As touched on in Section 6.3.1, in the presence of uncertainty and therefore existence of
many interest rates, the issue becomes more complicated.
11
Blanchard (2011).
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.4. Debt arithmetic
219
According to (6.11), key fundamentals (the spending- and tax-income ratios)
are consistent with absence of an explosive debt-income ratio as long as r is
unchanged. Nevertheless …nancial investors may be worried about default if b0
is high. Investors are aware that a rise in the actual interest rate, r; can always
happen and that if it does, a situation with r > r is looming, in particular if the
country has high debt. The larger is b0 ; the lower is the critical interest rate, r;
as witnessed by (6.12).
The worrying scenario is that the fear of default triggers a risk premium, and
if the resulting level of the interest rate on the debt, say r0 ; exceeds r, unpleasant
debt dynamics like that in the top panel of Fig. 6.2 set in. To r0 corresponds a
new value of the primary surplus, say s^0 ; de…ned by s^0 = (r0 gY )b0 : So s^0 is the
minimum primary surplus (as a share of GDP) required for a non-accelerating
debt-income ratio in the new situation. Since b0 > 0;
r0 > r ) (r
gY )b0 < (r0
gY )b0 ) s < s^0 ;
with s given in (6.11). The government could possibly increase its primary surplus, s; but at most up to s; and this will not be enough since the required primary
surplus, s^0 ; exceeds s: The situation would be as illustrated in the top panel of
Fig. 6. 2 with b given as s=(r0 gY ) < b0 :
That is, if the actual interest rate should rise above the critical interest rate,
r; runaway debt dynamics would take o¤ and debt default thereby be threatening.
A fear that it may happen may be enough to trigger a fall in the market price of
government bonds which means a rise in the actual interest rate, r. So …nancial
investors’fear can be a self-ful…lling prophesy. Moreover, as we saw in connection
with (6.12), the risk that r becomes greater than r is larger the larger is b0 .
For countries with high or low (low capability of collecting taxes as in
Greece for instance) a high b0 is therefore problematic. Across countries there is
no common threshold value for a “too large” public debt-to-income ratio. This
is because variables like ; ; r; and gY , as well as the net foreign debt position
and the current account de…cit, di¤er across countries. Late 2010 Greece had
(gross) government debt of 148 percent of GDP and the interest rate on 10-year
government bonds skyrocketed. Conversely Japan had (gross) government debt
of more than 200 percent of GDP while the interest rate on 10-year government
bonds remained very low.
Finer shades
1. As we have just seen, even when in a longer-run perspective a solvency problem
is unlikely, self-ful…lling expectations can here and now lead to default. Such a
situation is known as a liquidity crisis rather than a true solvency crisis. In a
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
220
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
liquidity crisis there is an acute problem of insu¢ cient cash to pay the next bill
on time (“cash-‡ow insolvency”) because lending is di¢ cult due to actual and
potential creditors’fear of default. A liquidity crisis can be braked by the central
bank stepping in and acting as a “lender of last resort” by printing money. In a
country with its own currency, the central bank can do so and thereby prevent a
bad self-ful…lling expectations equilibrium to unfold.12
2. In the above analysis we simpli…ed by assuming that several variables,
including ; ; and r; are constants. The upward trend in the old-age dependency
ratio, due to a decreased birth rate and rising life expectancy, together with a
rising request for medical care is likely to generate upward pressure on . Thereby
a high initial debt-income ratio becomes more challenging.
3. On the other hand, rBt is income to the private sector and can be taxed at
the same average tax rate as factor income, Yt : Then the benign inequality is
no longer r gY but (1
)r gY ; which is more likely to hold. Taxing interest
income is thus supportive of …scal sustainability (cf. Exercise B.28).
4. Having assumed a constant gY ; we have ignored business cycle ‡uctuations.
Allowing for booms and recessions, the timing of …scal consolidation in a country
with a structural primary surplus gap (s s^ > 0) becomes an important issue.
The case study in the next section will be an opportunity to touch upon this
issue.
6.4.2
Case study: The Stability and Growth Pact of the
EMU
The Maastrict criteria, after the Treaty of Maastrict 1992, for joining the Economic and Monetary Union (EMU) of the EU speci…ed both a government de…cit
rule and a government debt rule. The …rst is the rule saying that the nominal
budget de…cit must not be above 3 percent of nominal GDP. The debt rule says
that the debt should not be above 60 percent of GDP. The de…cit rule and the
debt rule were upheld in the Stability and Growth Pact (SGP) which was implemented in 1997 as the key …scal constituent of the EMU institutional framework.
Some of the EMU member states (Greece, Italy, and Belgium) have had debtincome ratios above 100 percent since the early 1990s. Yet they became full
12
In a monetary union which is not also a …scal union (think of the eurozone), the situation
is more complicated. A single member country with large government debt (or large debt in
large commercial banks for that matter) may …nd itself in an acute liquidity crisis without its
own means to solve it. We may interpret the elevation of interest rates on government bonds in
the Southern part of the eurozone in 2010-2012 as a manifestation of investors’fear of payment
di¢ culties. The elevation was not reversed until the European Central Bank in September 2012
declared its willingness to e¤ectively act as a “lender of last resort” (on a conditional basis),
see Box 6.3 in Section 6.4.2.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.4. Debt arithmetic
221
members of the EMU, vaguely committing to gradual reduction of their debtincome ratios. The 60 percent debt rule in the SGP is to be understood as a
long-run ceiling and, by the stock nature of debt, cannot be accomplished here
and now.
The two rules (with associated detailed contingencies and arrangements including ultimate pecuniary …nes for de…ance) are meant as discipline devices
aiming at “sound budgetary policy”, alternatively called “…scal prudence”. The
declared goal is protection of the European Central Bank (ECB) against political demands to loosen monetary policy in situations of …scal distress. A …scal
crisis in one or more of the eurozone countries, perhaps “too big to fail”, could
set in and entail a state of a¤airs approaching default on government debt and
chaos in the banking sector with rising interest rates spreading to other member
countries (a negative externality). Such a situation could generate open or concealed political pressure on the ECB to in‡ate away the real value of the debt
of the government in …scal crisis or to curb soaring interest rates and the risk of
contagion to neighboring countries through buying government bonds from the
country in trouble.
In fact, this scenario is what we saw in southern Europe in the wake of the
Great Recession triggered by the …nancial crisis starting late 2007. The lid on
de…cit spending imposed by the SGP was meant as a remedy to prevent such a
situation which might interfere with the ECB’s mandatory one and only concern
with “price stability”. The ECB interprets “price stability” as a consumer price
in‡ation rate “below, but close to, 2 percent per year over the medium term”.
The link between the de…cit and the debt rule
Whatever the virtues or vices of the SGP and whatever its future status, one may
ask the plain question: what is the arithmetical relationship, if any, between the
3 percent and 60 percent tenets?
First a remark about measurement. The measure of government debt, called
the EMU debt, used in the SGP criterion is based on the book value of the …nancial liabilities rather than the market value. In addition, the EMU debt is more
of a gross nature than the standard government net debt measure, corresponding
to our D. The EMU debt measure allows fewer of the government …nancial assets
to be subtracted from the government …nancial liabilities in order to reach a net
debt measure.13 In our calculation and subsequent discussion we ignore these
complications.
13
For Denmark the di¤erence between the two measures is substantial. In 2013 the Danish
EMU debt was 44.6% of GDP while the government net debt was 5.5% of GDP (Danish Ministry
of Finance, 2014).
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
222
Consider a de…cit rule saying that the (total) nominal budget de…cit must
never be above
100 percent of nominal GDP. By (6.3) with Mt+1 “small”,
so that it can here be ignored, this de…cit rule is equivalent to the requirement
Dt+1
Dt = GBDt = it Dt + Pt (Gt
Tt )
P t Yt :
(6.13)
In the SGP, = 0:03. Here we consider the general case:
> 0: To see the
implication for the (public) debt-to-income ratio in the long run, let us …rst
imagine a situation where the de…cit ceiling, ; is always binding for the economy
we look at. Then Dt+1 = Dt + Pt Yt and so
bt+1
Bt+1
Yt+1
Dt+1
Dt
=
+
;
Pt Yt+1
(1 + )Pt 1 (1 + gY )Yt 1 + gY
assuming constant output growth rate, gY ; and in‡ation rate : This reduces to
bt+1 =
1
bt +
:
(1 + )(1 + gY )
1 + gY
(6.14)
Assuming that (1 + )(1 + gY ) > 1 (as is normal over the medium run), this linear
di¤erence equation has the stable solution
bt = (b0
b)
1
(1 + )(1 + gY )
where
b =
t
+ b ! b for t ! 1;
(1 + )
(1 + )(1 + gY )
1
:
(6.15)
(6.16)
Consequently, if the de…cit rule (6.13) is always binding, the debt-income ratio
tends in the long run to be proportional to the de…cit bound : The factor of
proportionality is a decreasing function of the long-run growth rate of real GDP
and the in‡ation rate. This result con…rms the general tenet that if there is
economic growth, perpetual budget de…cits need not lead to …scal problems.
If on the other hand the de…cit rule is not always binding, then the budget
de…cit is on average smaller than above so that the debt-income ratio will in the
long run be smaller than b .
The conclusion is the following. With one year as the time unit, suppose the
de…cit rule is
= 0:03 and that gY = 0:03 and
= 0:02 (the upper end of
the in‡ation interval aimed at by the ECB). Suppose further the de…cit rule is
never violated. Then in the long run the debt-income ratio will be at most b
= 1:02 0:03=(1:02 1:03 1) 0:60: This is in agreement with the debt rule
of the SGP according to which the maximum value allowed for the debt-income
ratio is 60%.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.4. Debt arithmetic
223
Although there is nothing sacred about either of the numbers 0:60 or 0:03;
they are mutually consistent.
We observe that the de…cit rule (6.13) implies that:
The upper bound, b ; on the long-run debt income ratio is lower the higher
is in‡ation. The reason is that the growth factor
[(1 + )(1 + gY )] 1
for bt in (6.14) depends negatively on the in‡ation rate, . So does therefore
b since, by (6.15), b
(1 + gY ) 1 (1
) 1:
For a given ; the upper bound on the long-run debt income ratio is independent of both the nominal and real interest rate (this follows from the
indicated formula for the growth factor for bt and the fact that (1+i)(1+r) 1
= 1 + ):
The debate about the design of the SGP
By its design the SGP has short-run consequences for the economy in addition
to the long-run consequences. An examination and evaluation of of the SGP
cannot therefore ignore how the economy functions in the short run. The e¤ects
of changes in government spending and taxation depend much on the “state of
the business cycle”: is the economy in a boom with full capacity utilization or
in a slump with slack aggregate demand? Much of the debate about the SGP
has centered around the consequences of the de…cit rule in an economic recession triggered by a collapse of aggregate demand (for instance due to private
deleveraging in the wake of a banking sector crisis). Although the EMU member
countries are di¤erent economically, politically, and culturally, they are subject
to the same one-size-…ts-all monetary policy. Facing asymmetric shocks, the individual member countries in need of aggregate demand stimulation in a recession
have by joining the EMU renounced on both interest rate policy and currency
depreciation. The only policy tool left for demand stimulation is therefore …scal
policy. Instead of a supranational …scal authority responsible for handling the
problem, it is up to the individual member countries to act and to do so within
the constraints of the SGP.
On this background, among the criticisms put forward of the de…cit rule of
the SGP are the following. (The following discussion assumes acquaintance with
the elementary Keynesian income-expenditure model, where output is demanddetermined and below capacity, while the general price level is sticky.)
Criticisms 1. When considering the need for …scal stimuli in a recession, a
ceiling at 0:03 is too low unless the country has almost no government debt in
advance. Such a de…cit rule gives too little scope for counter-cyclical …scal policy,
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
224
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
including the free working of the automatic …scal stabilizers (i.e., the provisions,
through tax and transfer codes, in the government budget that automatically
cause tax revenues to fall and spending to rise when GDP falls). As an economy
moves towards recession, the de…cit rule may, bizarrely, force the government
to tighten …scal policy although the situation calls for a stimulation of aggregate demand. The pact has therefore sometimes been called the “Instability and
Depression Pact”. It imposes a wrong timing of …scal consolidation.14
2. There is little reason to have a ‡ow rule (a de…cit rule) on top of the stock
rule (the debt rule), since it is long-run …scal sustainability that really matters
for whether the …scal accounts are worrying. If one wants nevertheless to have a
supranational de…cit rule, it should be designed in a more ‡exible way than the
3% rule of the SGP. A meaningful de…cit rule would relate the de…cit to the trend
nominal GDP, which we may denote (P Y ) . Such a criterion would imply
GBD
(P Y ) :
(6.17)
Then
(P Y )
GBD
:
PY
PY
In recessions the ratio (P Y ) =(P Y ) is high, in booms it is low. This has the
advantage of allowing more room for budget de…cits when they are needed
without interfering with the long-run aim of stabilizing government debt below
some speci…ed ceiling.
3. A further step in this direction is a rule directly in terms of the structural
or cyclically adjusted budget de…cit rather than the actual year-by-year de…cit.
The cyclically adjusted budget de…cit in a given year is de…ned as the value the
de…cit would take in case actual output were equal to trend output in that year.
Denoting the cyclically adjusted budget de…cit GBD ; the rule would be
GBD
(P Y )
:
In fact, in its original version as of 1997 the SGP contained an additional rule
like that, but in the very strict form of
0: This requirement was implicit
in the directive that the cyclically adjusted budget “should be close to balance
14
The SGP has an exemption clause referring to “exceptional”circumstances. These circumstances were originally de…ned as “severe economic recession”, interpreted as an annual fall
in real GDP of at least 1-2%. By the reform of the SGP in March 2005, the interpretation
was changed into simply “negative growth”. Owing to the international economic crisis that
broke out in 2008, the de…cit rule was thus suspended in 2009 and 2010 for most of the EMU
countries. But the European Commission brought the rule into e¤ect again from 2011, which
according to many critics was much too early, given the circumstances.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.4. Debt arithmetic
225
or in surplus”. By this rule it is imposed that the debt-income ratio should be
close to zero in the long run. Many EMU countries certainly had
and have
larger cyclically adjusted de…cits. Taking steps to comply with such a low
structural de…cit ceiling may be hard and may bring national welfare down by
getting in the way of key tasks of the public sector. The minor reform of the
SGP endorsed in March 2005 allowed more contingencies, also concerning this
structural bound. And by the more recent reform, the Fiscal Pact of 2012, the
lid on the cyclically adjusted de…cit-income ratio was raised to 0.5%. This is still
a quite small number. Complying with this rule implies a long-run debt-income
ratio of at most 10%; given structural in‡ation and structural GDP growth at
2% and 3% per year, respectively.15
4. Regarding the composition of government expenditure, critics have argued
that the SGP pact entails a problematic disincentive for public investment. The
view is that a …scal rule should be based on a proper accounting of public investment instead of simply ignoring the composition of government expenditure. We
consider this issue in Section 6.6 below.
5. At a more general level critics have contended that bureaucratic budget
rules imposed on sovereign nations will never be able to do their job unless they
encompass stronger incentive-compatible elements.
Counterarguments Among the counterarguments raised against the criticisms
of the SGP has been that the potential bene…ts of the proposed alternative rules
are more than o¤set by the costs in terms of reduced simplicity, measurability,
and transparency (Schuknecht, 2005). The lack of ‡exibility may even be a good
thing because it helps “tying the hands of elected policy makers”. These points are
sometimes linked to a view that market economies are generally self-regulating.
Keynesian stabilization policy is not needed and may do more harm than good.
Indeed, in its original design the EMU was much in‡uenced by European followers
of Milton Friedman’s economic thinking named monetarism. (SOURCES)
Box 6.2. The 2010-2012 debt crisis in the Eurozone
What began as a banking crisis became a deep economic recession combined with a
government debt crisis.
At the end of 2009, in the aftermath of the global economic downturn, it became
evident that Greece faced an acute debt crisis driven by three factors: high government
debt, low ability to collect taxes, and lack of competitiveness due to cost in‡ation.
Anxiety broke out about the debt crisis spilling over to Spain, Portugal, Italy, and
15
The 10% follows from (6.16). In a historical perspective this is a very low long-run debtincome ratio for which there is little foundation in public economics or general welfare theory.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
226
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
Ireland, thus widening bond yield spreads in these countries vis-a-vis Germany in the
midst of a serious economic recession. Moreover, the solvency of big German banks
that were among the prime creditors of Greece was endangered. The major Eurozone
governments and the International Monetary Fund (IMF) reached an agreement to
help Greece (and indirectly its creditors) with loans and guarantees for loans, conditional on the government of Greece imposing yet another round of harsh …scal austerity
measures. The elevated bond interest rates of Greece, Italy, and Spain were not convincingly curbed, however, until in August-September 2012 the president of the ECB,
Mario Draghi, launched the “Outright Monetary Transactions” (OMT) program according to which, under certain conditions, the ECB will buy government bonds in
secondary bond markets with the aim of “safeguarding an appropriate monetary policy
transmission and the singleness of the monetary policy” and with “no ex ante quantitative limits”. Considerably reduced government bond spreads followed and so the
sheer announcement of the program seemed e¤ective in its own right. Doubts raised by
the German Constitutional Court about its legality vis-à-vis Treaties of the European
Union were …nally repudiated by the European Court of Justice mid-June 2015. At
the time of writing (late June 2015) the OMT program has not been used in practice.
Early 2015, a di¤erent massive program for purchases of government bonds, including
long-term bonds, in the secondary market as well as private asset-backed bonds was
decided and implemented by the ECB. The declared aim was to brake threatening de‡ation and return to “price stability”, by which is meant in‡ation close to 2 percent
per year.
So much about the monetary policy response. What about …scal policy? On the
basis of the SGP, the EU Commission imposed “…scal consolidation” initiatives to be
carried out in most EU countries in the period 2011-2013 (some of the countries were
required to start already in 2010). With what consequences? By many observers, partly
including the research department of IMF, the initiatives were judged self-defeating.
When at the same time comprehensive deleveraging in the private sector is going on,
“austerity” policy deteriorates aggregate demand further and raises unemployment.
Thereby, instead of budget de…cits being decreased, the numerator in the debt-income
ratio, D=(P Y ); is decreased: Fiscal multipliers are judged to be large (“in the 0.9 to
1.7 range since the Great Recession”, IMF, World Economic Outlook, Oct. 2012) in
a situation of idle resources, monetary policy aiming at low interest rates, and negative spillover e¤ects through trade linkages when “…scal consolidation”is synchronized
across countries. The unemployment rate in the eurozone countries was elevated from
7.5 percent in 2008 to 12 percent in 2013. The British economists, Holland and Portes
(2012), concluded: “It is ironic that, given that the EU was set up in part to avoid
coordination failures in economic policy, it should deliver the exact opposite”.
The whole crisis has pointed to a basic di¢ culty faced by the Eurozone. In spite
of the member countries being economically very di¤erent sovereign nations, they are
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.5. Solvency and the intertemporal government budget constraint
227
subordinate to the same one-size-…ts-all monetary policy without sharing a federal
government ready to use …scal instruments to mitigate regional consequences of adverse
asymmetric shocks. Bureaucratic policy rules and surveillance procedures for the …scal
policy of sovereign nations can hardly replace self-regulation; enforcement mechanisms
are bound to be weak. Moreover, abiding by the …scal rules of the SGP was certainly no
assurance of not ending up in a …scal crisis in the wake of a crisis in the banking sector,
as witnessed by Ireland and Spain. A seemingly robust …scal position can vaporize fast,
particularly if banks, “too big to fail”, need be bailed out. (END OF BOX)
6.5
Solvency and the intertemporal government
budget constraint
Up to now we have considered the issue of government solvency from the perspective of dynamics of the government debt-to-income ratio. It is sometimes useful
to view government solvency from another angle, that of the intertemporal budget constraint (GIBC). Under a certain condition stated below, the intertemporal
budget constraint is as relevant for a government as for private agents. A simple condition closely linked to whether the government’s intertemporal budget
constraint is satis…ed or not is what is known as the government’s No-PonziGame (NPG) condition. It is convenient to …rst consider this condition and its
relationship to government solvency (understood as absence of an accelerating
debt-income ratio). We concentrate on government net debt measured in real
terms.
6.5.1
When is the NPG condition necessary for solvency?
Consider a situation with a constant interest rate r: Suppose taxes are lump sum
or, at least, that there is no tax on interest income from owning government
bonds. Then the government’s NPG condition is that the present discounted
value of the public debt in the far future is not positive, i.e.,
lim Bt (1 + r)
t!1
t
0:
(NPG)
This condition says that government debt is not allowed to grow in the long
run at a rate as high as (or even higher than) the interest rate.16 That is, a
…scal policy satisfying the NPG condition rules out a permanent debt rollover.
Indeed, as we saw in Section 6.3.1, with B0 > 0; a permanent debt rollover
16
If there is e¤ective taxation of interest income at the rate r 2 (0; 1); then the aftertax interest rate, (1
r )r; is the relevant discount rate, and the NPG condition would read
t
limt!1 Bt [1 + (1
)r]
0:
r
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
228
policy (…nancing all interest payments and perhaps even also part of the primary
government spending) by debt issue leads to Bt B0 (1 + r)t for t = 0; 1; 2; : : : :
Substituting into (NPG) gives limt!1 Bt
B0 (1 + r)t (1 + r) t = B0 > 0; thus
violating (NPG).
The designation No-Ponzi-Game condition refers to a guy from Boston, Charles
Ponzi, who in the 1920s made a fortune out of an investment scam based on the
chain-letter principle. The principle is to pay o¤ old investors with money from
new investors. Ponzi was sentenced to many years in prison for his transactions;
he died poor and without friends.
True, this kind of …nancing behavior is not forbidden for the government as it
is typically for private agents. But under “normal” circumstances a government
has to plan its expenditures and taxation so as to comply with its NPG condition
since otherwise not enough lenders will be forthcoming.
Since the state is in principle in…nitely-lived, however, there is no …nal date
where all government debt should be over and done with. Indeed, the NPG
condition does not even require that the debt has ultimately to be non-increasing.
The NPG condition “only” says that the debtor, here the government, is not
allowed to let the debt grow forever at a rate as high as (or higher than) the
interest rate. For instance the U.K. as well as the U.S. governments have had
positive debt for centuries.
Suppose Y (GDP) grows at the constant rate gY (actually, for most of the
following results it would be enough that limt!1 Yt+1 =Yt = 1 + gY ). We have:
PROPOSITION 1 Let bt Bt =Yt and interpret “solvency”as absence of an for
ever accelerating debt-income ratio. Then:
(i) if r > gY ; solvency requires (NPG) satis…ed;
(ii) if r
gY ; the government can remain solvent without (NPG) being satis…ed.
Proof. When bt 6= 0;
lim
t!1
bt+1
bt
lim
t!1
Bt+1 =Yt+1
Bt+1 =Bt
Bt+1 =Bt
= lim
= lim
:
t!1
t!1
Bt =Yt
Yt+1 =Yt
1 + gY
(6.18)
Case (i): r > gY : If limt!1 Bt
0; then (NPG) is trivially satis…ed. Assume limt!1 Bt > 0: For this situation we prove the statement by contradiction. Suppose (NPG) is not satis…ed. Then, limt!1 Bt (1 + r) t > 0; implying
that limt!1 Bt+1 =Bt
1 + r: In view of (6.18) this implies that limt!1 bt+1 =bt
(1 + r)=(1 + gY ) > 1: Thus, bt ! 1; which violates solvency. By contradiction,
this proves that solvency implies (NPG) when r > gY .
Case (ii): r gY : Consider the permanent debt roll-over policy Tt = Gt for
all t
0; and assume B0 > 0. By (DGBC) of Section 6.2 this policy yields
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.5. Solvency and the intertemporal government budget constraint
229
Bt+1 =Bt = 1 + r; hence, in view of (6.18), lim t!1 bt+1 =bt = (1 + r)=(1 + gY )
1: The policy consequently implies solvency. On the other hand the solution
of the di¤erence equation Bt+1 = (1 + r)Bt is Bt = B0 (1 + r)t : Thus Bt (1 + r) t
= B0 > 0 for all t; thus violating (NPG).
Hence imposition of the NPG condition on the government relies on the interest rate being in the long run higher than the growth rate of GDP. If instead
r
gY ; the government can cut taxes, run a budget de…cit, and postpone the
tax burden inde…nitely. So in that case the government can run a Ponzi Game
and still stay solvent. Nevertheless, as alluded to earlier, if uncertainty is added
to the picture, there will be many di¤erent interest rates, matters become more
complicated, and quali…cations to Proposition 1 are needed (Blanchard and Weil,
2001). The prevalent view among macroeconomists is that imposition of the NPG
condition on the government is generally warranted.
While in the case r > gY ; the NPG condition is necessary for solvency, it is
not su¢ cient. Indeed, we could have
1 + gY < lim Bt+1 =Bt < 1 + r:
t!1
(6.19)
Here, by the upper inequality, (NPG) is satis…ed, yet, by the lower inequality
together with (6.18), we have limt!1 bt+1 =bt > 1 so that the debt-income ratio
explodes.
EXAMPLE 1 Let GDP = Y; a constant, and r > 0; so r > gY = 0: Let the
budget de…cit in real terms equal "Bt + ; where 0 " < r and > 0: Assuming
no money-…nancing of the de…cit, government debt evolves according to Bt+1 Bt
= "Bt + which implies a simple linear di¤erence equation:
Bt+1 = (1 + ")Bt + :
(*)
Case 1: " = 0: Then the solution of (*) is
Bt = B0 + t;
(**)
B0 being historically given. Then Bt (1 + r) t = B0 (1 + r) t + t(1 + r) t ! 0 for
t ! 1: So, (NPG) is satis…ed. Yet the debt-GDP ratio, Bt =Y; goes to in…nity
for t ! 1: That is, in spite of (NPG) being satis…ed, solvency is not present. For
" = 0 we thus get the insolvency result even though the lower strict inequality in
(6.19) is not satis…ed. Indeed, (**) implies Bt+1 =Bt = 1 + =Bt ! 1 for t ! 1
and 1 + gY = 1:
Case 2: 0 < " < r: Then the solution of (*) is
Bt = (B0 +
"
)(1 + ")t
"
! 1 for t ! 1;
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
230
if B0 >
=": So Bt =Y ! 1 for t ! 1 and solvency is violated. Nevertheless
Bt (1 + r) t ! 0 for t ! 1 so that (NPG) holds.
The example of this case fully complies with both strict inequalities in (6.19)
because Bt+1 =Bt = 1 + " + =Bt ! 1 + " for t ! 1:
An approach to …scal budgeting that ensures debt stabilization and thereby
solvency is the following. First, impose that the cyclically adjusted primary
budget surplus as a share of GDP equals a constant, s. Next, adjust taxes and/or
spending such that s
s^ = (r gY )b0 , ignoring short-run di¤erences between
Yt+1 =Yt and 1 + gY and between rt and its long-run value, r; as in (6.10), s^ is the
minimum primary surplus as a share of GDP required to obtain bt+1 =bt
1 for
all t 0. This s^ is a measure of the burden that the government debt imposes
on the tax payers. If the policy steps needed to realize at least s^ are not taken,
the debt-income ratio will grow, thus worsening the …scal position in the future
by increasing s^.
6.5.2
Equivalence of NPG and GIBC
The condition under which the NPG condition is relevant for solvency is also
the condition under which the government’s intertemporal budget constraint is
relevant. To show this we let t denote the current period and t + i denote a
period in the future. As above, we ignore seigniorage. Debt accumulation is then
described by
Bt+1 = (1 + r)Bt + Gt + Xt
T~t ;
where Bt is given.
(6.20)
The government intertemporal budget constraint (GIBC), as seen from the beginning of period t; is the requirement
1
X
i=0
(Gt+i + Xt+i )(1 + r)
(i+1)
1
X
T~t+i (1 + r)
(i+1)
Bt :
(GIBC)
i=0
This condition requires that the present value of current and expected future
government spending does not exceed the government’s net wealth (the present
value of currentP
and expected future tax revenue
minus existing government debt).
PI
By the symbol 1
x
we
mean
lim
x
I!1
i=0 i
i=0 i : Until further notice we assume
this limit exists.
To see the connection between the dynamic accounting relationship (6.20) and
the intertemporal budget constraint, note that by rearranging (6.20) and using
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.5. Solvency and the intertemporal government budget constraint
231
forward substitution we get
Bt = (1 + r) 1 (T~t Xt Gt ) + (1 + r) 1 Bt+1
j
X
(1 + r) (i+1) (T~t+i Xt+i Gt+i ) + (1 + r)
=
=
i=0
1
X
(1 + r)
(i+1)
(1 + r)
(i+1)
(T~t+i
Xt+i
Gt+i ) + lim (1 + r)
(T~t+i
Xt+i
Gt+i );
Bt+j+1
(j+1)
j!1
i=0
1
X
(j+1)
Bt+j+1
(6.21)
i=0
if and only if government debt ultimately grows at a rate less than r so that
lim (1 + r)
j!1
(j+1)
Bt+j+1
0:
(6.22)
This latter condition is exactly the NPG condition above (replace t in (6.22) by
0 and j by t 1). And the condition (6.21) is just a rewriting of (GIBC). We
conclude:
PROPOSITION 2 Given the book-keeping relation (6.20), then:
(i) (NPG) is satis…ed if and only if (GIBC) is satis…ed;
(ii) there is strict equality in (NPG) if and only if there is strict equality in
(GIBC).
We know from Proposition 1 that in the “normal case”where r > gY ; (NPG) is
needed for government solvency. The message of (i) of Proposition 2 is that then
also (GIBC) need be satis…ed. Given r > gY ; to appear solvent a government thus
has to realistically plan taxation and spending pro…les such that the present value
of current and expected future primary budget surpluses matches the current
debt, cf. (6.21). Otherwise debt default is looming and forward-looking investors
will refuse to buy government bonds or only buy them at a reduced price, thereby
aggravating the …scal conditions.17
In view of the remarks around the inequalities in (6.19), however, satisfying
the condition (6.21) is only a necessary condition (if r > gY ), not in itself a
su¢ cient condition for solvency. A simple condition under which satisfying the
17
Government debt defaults have their own economic as well as political costs. Yet, they
occur now and then. Recent examples include Russia in 1998 and Argentina in 2001-2002.
During 2010-12, Greece was on the brink of debt default.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
232
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
(6.21) is su¢ cient for solvency is that both Gt and Tt are proportional to Yt ; cf.
Example 2.
EXAMPLE 2 Consider a small open economy facing an exogenous constant
real interest rate r: Suppose that at time t government debt is Bt > 0; GDP is
growing at the constant rate gY ; and r > gY : Assume Gt = Yt and Tt T~t Xt
= Yt ; where and are positive constants. What is the minimum size of the
primary budget surplus as a share of GDP required for satisfying the government’s
intertemporal budget constraint asPseen from time t? Inserting into the formula
(i+1)
(6.21), with strict equality, yields 1
(
)Yt+i = Bt : This gives
i=0 (1 + r)
P1 1+gY (i+1)
= r gY Yt = Bt ; where we have used the rule for the sum
Y
i=0
1+gY t
1+r
of an in…nite geometric series. We conclude that
= (r
gY )
Bt
:
Yt
This is the same result as in (6.10) above if we substitute s^ =
and t = 0.
Thus, satisfying the government’s intertemporal budget constraint ensures in this
example a constant debt-income ratio and thereby government solvency.
On the other hand, if r gY ; it follows from propositions 1 and 2 together that
the government can remain solvent without satisfying its intertemporal budget
constraint (at least as long as we ignore uncertainty). The background for this
fact may become more apparent when we recognize how the condition r
gY
a¤ects the constraint (GIBC). Indeed, to the extent that the tax revenue tends
to grow at the same rate as national income, we have T~t+i = T~t (1 + gY )i . By the
rule for the sum of an in…nite geometric series, this implies
1
X
i=0
T~t+i (1 + r)
(i+1)
1
T~t X
=
1 + gY i=0
1 + gY
1+r
(i+1)
;
whicch is in…nte if r gY : The PV of future tax revenues is thus unbounded in
this case. Suppose that also government spending, Gt+i + Xt+i ; grows at the rate
gY . Then the evolution of the primary surplus is described by T~t+i Xt+i Gt+i
= (T~t (Gt +Xt ))(1+gY )i : Although in this case also the PV of future government
spending is in…nite, (6.21) shows that any positive initial primary budget surplus,
T~t (Gt + Xt ); ever so small can repay any level of initial debt in …nite time.
In (GIBC) and (6.22) we allow strict inequalities to obtain. What is the
interpretation of strict inequality here? The answer is here:
COROLLARY OF PROPOSITION 2 Given the book-keeping relation (6.20),
then strict inequality in (GIBC) is equivalent to the government in the long run
accumulating positive net …nancial wealth.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.5. Solvency and the intertemporal government budget constraint
233
Proof. Strict inequality in (GIBC) is equivalent to strict inequality in (6.21),
which in turn, by (ii) of Proposition 2, is equivalent to strict inequality in (6.22),
which is equivalent to limj!1 (1 + r) (j+1) Bt+j+1 < 0: This latter inequality is
equivalent to limj!1 Bt+j+1 < 0, as, by de…nition, r > 1 so that limj!1 (1 +
r) (j+1) 0:
It is common to consider as the regular case the case where the government
does not attempt to accumulate positive net …nancial wealth in the long run and
so becoming a net creditor vis-à-vis the private sector. Returning to the condition
r > gY ; in the regular case …scal solvency thus amounts to the requirement
1
X
T~t+i (1 + r)
(i+1)
i=0
1
X
=
(Gt+i + Xt+i )(1 + r)
(i+1)
+ Bt ;
(GIBC’)
i=0
which is obtained by rearranging (GIBC) and replacing weak inequality with strict
equality. It is certainly not required that the budget is balanced all the time. The
point is “only” that for a given planned expenditure path, a government should
plan realistically a stream of future tax revenues the PV of which matches the
PV of planned expenditure plus the current debt. If an unplanned budget de…cit
is run so that the public debt rises during a recession, say then higher taxes
than otherwise must be levied in the future.
Or, rewriting (GIBC’) as
1
X
T~t+i
(Gt+i + Xt+i ) (1 + r)
(i+1)
= Bt ;
i=0
we see the basic principle that the present discounted value of planned future
primary surplusses must equal the initial debt. If debt is positive today, then the
government has to run a positive primary surplus for a su¢ ciently long time in
the future.
Finer shades
1. If the real interest rate varies over time, all the above formulas remain valid if
(1 + r) (i+1) is replaced by ij=0 (1 + rt+j ) 1 :
2. We have essentially ignored seigniorage. Under “normal” circumstances
seigniorage is present which relaxes (GIBC’) a little. Indeed, the monetary base,
M; typically grows at a rate at least as high as the GNP growth rate if higher
in the long run, then in‡ation tends to arise.
A simple theory says that the long-run in‡ation rate, ; tends to equal the
excess of the money growth rate, gM ; over the output growth rate, gY ; i.e.,
= gM gY :
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
234
.... (to be continued)
3. The case of a liquidity trap. Buiter 2014. Quantitative easing: refer to Ch.
33-2014.
6.6
A proper accounting of public investment
....
Proposal about …scal policy committees.
Moreover, in economic recessions government borrowing is usually cheap. As
Harvard economist Lawrence Summers put it: “Idle workers + Low interest rates
= Time to rebuild infrastructure”(Summers, 2014).
6.7
Ricardian equivalence?
We now turn to the question how budget policy a¤ects resource allocation and
intergenerational distribution. The role of budget policy for economic activity
within a time horizon corresponding to the business cycle is not the issue here.
Our question is about the longer run: does it matter for aggregate consumption
and aggregate saving in an economy with full capacity utilization whether the
government …nances its current spending by (lump-sum) taxes or borrowing?
6.7.1
Two di¤ering views
There are two opposite answers in the literature to this question. Some macroeconomists answer the question in the negative. This is the debt neutrality view,
also called the Ricardian equivalence view. The in‡uential American economist
Robert Barro is in this camp. Other macroeconomists answer the question in the
positive. This is the debt non-neutrality view or absence of Ricardian equivalence
view. The in‡uential French-American economist Olivier Blanchard is in this
camp.
The two di¤erent views rest on two di¤erent models of the economic reality.
The two models have a common point of departure, though, namely a situation
where:
1. r > gY ;
2. …scal policy satis…es the intertemporal budget constraint
1
X
t=0
T~t (1 + r)
(t+1)
1
X
=
(Gt + Xt )(1 + r)
t=0
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
(t+1)
+ B0 ;
(6.23)
6.7. Ricardian equivalence?
235
where the initial debt, B0 ; and the planned path of Gt + Xt are given;
3. agents have rational (model consistent) expectations;
4. some of the taxes are lump sum and only these are varied in the thought
experiment to be considered;
5. credit market imperfections are absent.
For a given planned time path of Gt + Xt ; equation (6.23) implies that a tax
cut in any period has to be met by an increase in future taxes of the same present
discounted value as the tax cut.
Ricardian equivalence
The Ricardian equivalence view is the conception that government debt is neutral
in the sense that for a given time path of government spending, aggregate private
consumption is una¤ected by a temporary tax cut. The temporary tax cut does
not make the households feel richer because they expect that the ensuing rise
in government debt will lead to higher taxes in the future. The essential claim
is that the timing of (lump-sum) taxes does not matter. The name Ricardian
equivalence comes from a seemingly false association of this view with the
early nineteenth-century British economist David Ricardo. It is true that Ricardo
articulated the possible logic behind debt neutrality. But he suggested several
reasons that debt neutrality would not hold in practice and in fact he warned
against high public debt levels (Ricardo, 1969, pp. 161-164). Therefore it is
doubtful whether Ricardo was a Ricardian.
Debt neutrality was rejuvenated, however, by Robert Barro in a paper entitled
“Are government bonds net wealth [of the private sector]?”, a question which
Barro answered in the negative (Barro 1974). Barro’s debt neutrality view rests
on a representative agent model, that is, a model where the household sector
is described as consisting of a …xed number of in…nitely-lived forward-looking
“dynasties”. With perfect …nancial markets, a change in the timing of taxes
does not change the present value of the in…nite stream of taxes imposed on the
individual dynasty. A cut in current taxes is o¤set by the expected higher future
taxes. Though current government saving (T G rB) goes down, private saving
and bequests left to the members of the next generation go up equally much.
More precisely, the logic of the debt neutrality view is as follows. Suppose,
for simplicity, that the government waits only 1 period to increase taxes and
then does so in one stroke. Then, for each unit of account current taxes are
reduced, taxes next period are increased by (1 + r) units of account. The present
value as seen from the end of the current period of this future tax increase is
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
236
(1 + r)=(1 + r) = 1: As 1 1 = 0; the change in the time pro…le of taxation will
make the dynasty feel neither richer nor poorer. Consequently, its current and
planned future consumption will be una¤ected. That is, its current saving goes up
just as much as its current taxation is reduced. In this way the altruistic parents
make sure that the next generation is fully compensated for the higher future
taxes. Current private consumption in society is thus una¤ected and aggregate
saving stays the same.18
Absence of Ricardian equivalence
Olivier Blanchard and others dissociate themselves from such representative agent
models because of their coarse description of the household sector. Instead attention is drawn to overlapping generations models which emphasize …nite lifetime
and life-cycle behavior of human beings and lead to a refutation of Ricardian
equivalence. The essential point is that those individuals who bene…t from lower
taxes today will at most be a fraction of those who bear the higher tax burden in
the future. As taxes levied at di¤erent times are thereby levied at partly di¤erent
sets of agents, the timing of taxes generally matters. The current tax cut makes
current tax payers feel wealthier and this tends to increase their consumption
and lead to a decrease in aggregate saving. The present generations bene…t and
future tax payers (partly future generations) bear the cost in the form of access
to less national wealth than otherwise.
The next subsection provides an example showing in detail how a change
in the timing of taxes a¤ects aggregate private consumption in an overlapping
generations framework.
6.7.2
A small open OLG economy with a temporary budget de…cit
We consider a Diamond-style overlapping generations (OLG) model of a small
open economy (henceforth named SOE) with a government sector. The relationship between SOE and international markets is described by the same four
assumptions as in Section 5.3 of Chapter 5:
(a) There is perfect mobility of goods and …nancial capital across borders.
(b) There is no uncertainty and domestic and foreign …nancial claims are perfect
substitutes.
18
The complete Barro model is presented in Chapter 7.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.7. Ricardian equivalence?
237
(c) The need for means of payment is ignored; hence so is the need for a foreign
exchange market.
(d) There is no labor mobility across borders.
The assumptions (a) and (b) imply real interest rate equality. That is, in
equilibrium the real interest rate in SOE must equal the real interest rate in
the world …nancial market, r. And by saying that SOE is “small” we mean
it is small enough to not a¤ect the world market interest rate as well as other
world market factors. We imagine that all countries trade one and the same
homogeneous good. International trade will then only be intertemporal trade,
i.e., international borrowing and lending of this good.
We assume that r is constant over time and that r > n
0. As earlier
we let Lt denote the size of the young generation and Lt = L0 (1 + n)t . Each
young supplies one unit of labor inelastically, hence Lt is aggregate labor supply.
Assuming full employment, gross domestic product, GDP, is Yt = F (Kt ; Lt ):
Some national accounting for the open economy
Gross national saving is
S t = Yt
rN F Dt
Ct
Gt = Yt
rN F Dt
(c1t Lt + c2t Lt 1 )
Gt ;
(6.24)
where N F Dt is (net) foreign debt (also called external debt) at the beginning
of period t; Gt is government consumption in period t; and c1t and c2t are consumption by a young in period t and an old in period t; respectively. In the open
economy, generally, gross investment, It ; di¤ers from gross saving. If N F Dt > 0;
the interpretation is that some of the capital stock, Kt ; is directly or indirectly
owned by foreigners. On the other hand, if N F Dt < 0; SOE has positive net
claims on resources in the rest of the world.
National wealth, Vt ; of SOE at the beginning of period t is, by de…nition,
national assets minus national liabilities,
Vt
Kt
N F Dt :
National wealth is also, by de…nition, the sum of private …nancial (net) wealth,
At ; and government …nancial (net) wealth, Bt , Bt being government (net) debt
(we assume the government has no physical assets). Thus,
Vt
At + ( Bt ):
(6.25)
When the young save, they accumulate private …nancial wealth. The private
…nancial wealth at the start of period t+1 must in our Diamond framework equal
the (net) saving by the young in the previous period, i.e.,
At+1 = st Lt
N
S1t
:
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
(6.26)
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
238
So, by (6.25) forwarded one period, we have
Vt+1 = At+1 + ( Bt+1 ) = st Lt
(6.27)
Bt+1 :
Remark. The reader might alternatively want to reach this conclusion from the
perspective of national saving. By de…nition the increase in national wealth
equals net national saving StN = St Kt ; which in turn equals net private saving,
st Lt +( At ); minus net public dissaving. The latter equals the government budget
de…cit, GBDt : That is,
Vt+1
N
N
N
Vt = StN = S1t
+ S2t
GBDt = S1t
+ ( At )
= At+1 At (Bt+1 Bt );
GBDt
N
is aggregate (net) saving of the old, and the last equality comes from
where S2t
(6.26) and the maintained assumption that budget de…cits are fully …nanced by
debt issue.
Firms’behavior
GDP is produced by an aggregate neoclassical production function with CRS:
Yt = F (Kt ; Lt ) = Lt F (kt ; 1)
Lt f (kt );
where Kt and Lt are input of capital and labor, respectively, and kt
Kt =Lt .
Technological change is ignored. Imposing perfect competition in all markets,
markets clear so that Lt can be interpreted as both employment and labor supply
(exogenous). Pro…t maximization leads to f 0 (kt ) = r + ; where is a constant
capital depreciation rate, 0
1. When f satis…es the condition limk!0 f 0 (k)
0
> r + > limk!1 f (k), there is always a solution in k to this equation and it is
unique (since f 00 < 0) and constant over time (as long as r and are constant):
Thus,
kt = f 0 1 (r + ) k; for all t:
(6.28)
The stock of capital, Kt ; is determined as Kt = kLt :
In view of …rms’pro…t maximization, the equilibrium real wage before tax is
wt =
@Yt
= f (k)
@Lt
f 0 (k)k
w;
a constant. GDP will evolve according to
Yt = f (k)Lt = f (k)L0 (1 + n)t = Y0 (1 + n)t = Y0 (1 + n)t
and thus grow at the same rate as the labor force, i.e., gY = n.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
(6.29)
6.7. Ricardian equivalence?
239
Government and household behavior
We assume that the role of the government sector is to deliver a service, a “public
good”. Let Gt be the size of this service in period t. Think of a non-rival good like
“rule of law”, TV-transmitted theatre, or another public service free of charge.
Suppose
Gt = G0 (1 + n)t ;
where 0 < G0 < F (K0 ; L0 ): It is assumed that the production of Gt uses the
same technology and therefore involves the same unit production costs as the
other components of GDP. As the focus is not on distortionary e¤ects of taxation,
taxes are assumed to be lump sum, i.e., levied on individuals irrespective of their
economic behavior.
Without consequences for the qualitative results in which we are interested,
we assume a CRRA period utility function u(c) = (c1
1)=(1
); where
> 0: To keep things simple, the utility of the public good enters additively in
life-time utility so that it does not a¤ect marginal utilities of private consumption.
In addition we assume that the public good does not a¤ect productivity in the
private sector. There is a tax on the young as well as the old in period t, 1 and
2 ; respectively; until further notice these are time-independent. Possibly, 1 or
2 is negative, in which case there is a transfer to either the young or the old.
The consumption-saving decision of the young will be the solution to the
following problem:
c11t
1
+ v(Gt ) + (1 + )
1
= w
1;
= (1 + r)st
2;
0; c1t+1 0;
max U (c1t ; c2t+1 ) =
c1t + st
c2t+1
c1t
1
c12t+1
1
1
+ v(Gt+1 )
s.t.
where the function v( ) represents the utility contribution of the public good. The
implied Euler equation can be written
c2t+1
=
c1t
1+r
1+
1=
:
Inserting the two budget constraints and solving for st , we get
st =
w
1
+
1 + (1 + )
1+ 1=
1+r
( 1)=
1+r
1+
2
s(w; r;
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
1;
2 ):
(6.30)
240
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
Consumption in the …rst and the second period then is
c1t = w
1
(6.31)
st = c^1 (r)ht
and
(6.32)
c2t+1 = c^2 (r)ht ;
respectively, where
1+
c^1 (r)
1+r
1+
1+ +
c^2 (r) =
(1
)=
2 (0; 1) and
1=
1+r
1+
(6.33)
1+r
c^1 (r) =
1 + (1 + )
1+r
1+
(
1)=
(6.34)
are the (average = marginal) propensities to consume out of wealth, and where
ht = w
2
1
1+r
h;
(6.35)
is the after-tax human wealth (the “endowment”) of the young, that is, the
present value, evaluated at the end of period t; of disposable lifetime income (the
“endowment”); we assume 1 and 2 are such that h > 0: Given r; individual
consumption in both the …rst and the second period of life is thus proportional
to individual human wealth. This is as expected in view of the homothetic life
time utility function. And if = r; then c^1 (r) = c^2 (r) = (1 + r)=(2 + r); i.e., there
is complete consumption smoothing as also the Euler equation indicates when
= r.19
The tax revenue in period t is Tt = 1 Lt + 2 Lt 1 = ( 1 + 2 =(1 + n))Lt :
Suppose B0 = 0 and that along the reference path the budget is and remains
balanced for all t; i.e., Tt = Gt = G0 (1 + n)t : Then the tax code ( 1 ; 2 ) must
satisfy
1
+
2
1+n
L0 = G0 :
Consistency with h > 0 in (6.35) requires a “not too large”G0 :
Along the reference path aggregate private consumption grows at the same
constant rate as GDP and public consumption, the rate n. Indeed,
Ct = c1t Lt +
c2t
c2t
Lt = (c1t +
)L0 (1 + n)t = C0 (1 + n)t :
1+n
1+n
19
By calculating backwards from (6.34) to (6.33) to (6.30), the reader may check whether the
calculated st ; c1t and c2t+1 are consistent.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.7. Ricardian equivalence?
241
A one-o¤ tax cut
As an alternative to the reference path, we now assume than an unexpected oneo¤ cut in taxation by x; 0 < x < 1 ; takes place in period 0 for every individual,
whether young or old. What are the consequences of this? The tax cut amounts
to creating a budget de…cit in period 0 equal to (L0 + L 1 )x: At the start of
period 1 there is thus a government debt B10 = (L0 + L 1 )x, while in the reference
path, B1 = 0. Since we assumed r > n = gY ; government solvency requires that
the present value of future taxes, as seen from the beginning of period 1, rises
by (L0 + L 1 )x. This may be accomplished by, for instance, raising the tax on
all individuals from period 1 onward by m. Suppose this way of addressing the
arisen debt is already in period 0 credibly announced by the government to be
followed. The required value of m will satisfy
1
X
(L0 + L 1 )(1 + n)t m(1 + r)
t
= (L0 + L 1 )x:
t=1
This gives
m
1
X
t=1
1+n
1+r
t
= x:
As r > n; applying the rule for the sum of an in…nite geometric series gives
m=
r n
x
1+n
m:
(6.36)
The higher is the interest rate r; the higher the needed rise in future taxes.
This is because the interest burden of the debt is higher. On the other hand, a
higher population growth rate, n; reduces the needed rise in future taxes. This is
because the interest burden per capita is mitigated by population growth. Finally,
a greater tax cut, x; in the …rst period implies greater tax rises in future periods.
Let the value of the variables along this alternative path be marked with a
prime. In period 0 the tax cut unambiguously bene…ts the old whose increase in
consumption equals the saved tax:
c020
c20 = x > 0:
(6.37)
The young in period 0 know that per capita taxes next period will be increased
by m: In view of the tax cut in period 0, the young nevertheless experiences an
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
242
increase in after-tax human wealth equal to
h00
+m
2
w
1
1+r
1+r
r n
x
(by (6.36))
=
1
(1 + r)(1 + n)
1 + (2 + r)n
x > 0:
=
(1 + r)(1 + n)
h0 = w
1
+x
2
Consequently, through the wealth e¤ect this generation enjoys increases in consumption through life equal to
c010
c021
c10 = c^1 (r)(h00
c21 = c^2 (r)(h00
(6.38)
(6.39)
h0 ) > 0;
h0 ) > 0;
by (6.31) and (6.32), respectively. So period t’s old generation and young generation bene…t from the temporary budget de…cit. But all future generations are
worse o¤ in that they do not bene…t from the tax relief in period 0, but instead
have to bear the future cost of the tax relief by a reduction in individual after-tax
human wealth. Indeed, for t = 1; 2; : : : ;
h0t
ht = h01
=
h=w
m+
m
1+r
1
m
+m
1+r
2
< 0:
w
2
1
1+r
(6.40)
All things considered, since both the young and the old in period 0 increase
their consumption, aggregate consumption in period 0 rises. Ricardian equivalence thus fails.
National saving and wealth accumulation*
The direct impact on national wealth of the temporary tax cut How
N
N
does aggregate private net saving, S10
+ S20
; respond to the temporary tax cut?
In both the reference path and the alternative path, the old enter period 0 with
the …nancial wealth A0 and leave the period with zero …nancial wealth. So their
N
net saving is S20
= A0 in both …scal regimes. Although the young in period 0
increase their consumption in response to the temporary tax cut, they increase
their period 0-saving as well. The increased saving by the young is revealed by
the fact that they in period 1, as old, can a¤ord to increase their consumption
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.7. Ricardian equivalence?
243
in spite of the tax increase of size m in that period. Indeed, from (6.39) and the
period budget constraint as old follows
0 < c021 c21 = (1 + r)s00 ( 2 + m) ((1 + r)s0
= (1 + r)(s00 s0 ) m < (1 + r)(s00 s0 );
2)
thus implying s00 s0 > 0: In that A01 =L0 = s00 > s0 = A1 =L0 ; also aggregate
private …nancial wealth per old at the beginning of period 1 is larger than it
would have been without the temporary tax cut. This might seem paradoxical in
view of the higher aggregate private consumption in period 0. The explanation
lies in the fact that the lower taxation in period 0 means higher disposable income,
allowing both higher private consumption and higher private saving in period 0.
Nevertheless, gross national saving, cf. (6.24), is lower than in the reference
path; indeed, C00 > C0 implies
S00 = F (K0 ; L0 )
rN F D0
C00
G0 < F (K0 ; L0 )
rN F D0
C0
G0 = S0 :
A counterpart of the increased private saving is the public dissaving, re‡ecting the
budget de…cit created one-to-one by the reduction in taxation. As the increased
disposable income resulting from the latter partly goes to increased private saving and partly to increased private consumption, the rise in private saving is
smaller than the public dissaving, gross national saving ends up lower than in
the reference path.
K0 : The public
Net national saving in the reference path is S0N = S0
dissaving in the alternative path reduces net national saving by the amount
S0N
S0N 0 = C00 C0 = c010 L0 + c020 L 1 (c10 L0 + c20 L 1 )
= (c010 c10 )L0 + (c020 c20 )L 1 = c^1 (r)(h00 h0 )L0 + xL 1
1
1 + (2 + r)n
xL0 + x
L0
= c^1 (r)
(1 + r)(1 + n)
1+n
1 + (2 + r)n
1
=
c^1 (r)
+1
L0 x > 0:
(6.41)
1+r
1+n
For our national income accounting to be consistent, national wealth should
decrease by the same amount as net national saving . Let us check. By the
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
244
de…nition (6.25) follows
V10 = A01
B10 = s00 L0
= (w
(1 +
c^1 (r)h00 ) L0
1
=
w
1
c^1 (r) w
=
w
1
c^1 (r) w
= s0 L0
= s0 L0
1
)L0 x = (w
1+n
1
L0 x
1+n
2+m
1+x
1+r
2
(
c010
x)L0
1
L0 x
1+n
L0
1
L0 x
1+n
1
L0 x
1+n
m
c^1 (r) x
1+r
1
L0 x
1+n
L0
L0
1
x)
1
1+r
r n
c^1 (r) 1
L0 x
(1 + r)(1 + n)
1 + (2 + r)n
1
c^1 (r)
+1
L0 x < s0 L0 = V1 :
1+r
1+n
(6.42)
We see that national wealth has decreased by an amount equal to the decrease
in net national saving in (6.41), as it should.
Future consequences As revealed by (6.40), all future generations (those born
in period 1; 2; : : : ) are worse o¤ along the alternative path. One might think that
also aggregate private …nancial wealth per old along the alternative path would
necessarily be lower. But this is not so. As of period t = 2; 3;. . . , aggregate
private …nancial wealth per old along the alternative path is
A0t =Lt
1
= s0t
1
= w
=w
(
= w
1
1
(
1
+ m)
m
c^1 (r)(w
c^1 (r)(w
= w
1
c^1 (r) w
1
=w
c^1 (r)
1)
1
(
m
1
1+r
)
2+r
1+r
+ m) c^1 (r)h0t
2+m
1+r
2
1+r
+ c^1 (r)
m
1+r
m
1+r
1
c^1 (r) 1 +
1+r
m + c^1 (r)m + c^1 (r)
2
1
1
+ c^1 (r)m + c^1 (r)
2
1
1
1
c^1 (r) w
= w
= st
c01t
+ m)
1+r
r n
x:
1+n
1
m
(6.43)
Thus, for t = 2; 3; : : : ;
A0t
At
Q
holds for s0t 1 Q st 1 ; respectively, which in turn holds for
Lt 1
Lt 1
1+r
; respectively.
(6.44)
c^1 (r) Q
2+r
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.7. Ricardian equivalence?
245
In the benchmark case = 1, (6.33) gives c^1 (r) = (1 + r)=(2 + ): In combination
with (6.44), this implies that aggregate private …nancial wealth per old along the
alternative path is lower than, equal to, or higher than along the reference path if
R r; respectively (in the benchmark case). The reason that it may be higher is
that the saving by the young, which next period constitutes the private …nancial
wealth, has to cover not only the consumption as old but also the taxes as old
which have been increased. In view of st = (c2t+1 + 2 )=(1 + r); a rise in 2 thus
gives scope for a rise in st at the same time as c2t+1 decreases.
What is sure, however, is that national wealth as of period t = 2; 3; : : : ; is
smaller along the alternative path. In Exercise 6.? the reader is asked to show
that for t = 1; 2; : : : ; Bt0 = [(2 + n)=(1 + n)] L1 (1 + n)t 2 x: With this evolution in
public debt, the evolution in national wealth per old along the alternative path
as of period t = 2; 3; : : : , is
Vt0
Lt 1
A0t
Lt 1
= st
= s0
< s0
< s0
Bt
= s0t
Lt 1
2+n
x
1+n
2+r
1
1 c^1 (r)
(r n) + 2 + n
x
(from (6.43))
1
1+r
1+n
1
1+r
1 + (2 + r)n
+1
x (1 c^1 (r))
x
c^1 (r)
1+r
1+n
1+n
1 + (2 + r)n
1
c^1 (r)
+1
x = V10 =L0
(by (6.42))
1+r
1+n
At
Vt
A1
=
=
;
=
L0
Lt 1
Lt 1
1
where the second to last inequality is due to c^1 (r) < 1; cf. (6.33), while the
two …rst equalities in the last line are due to the constancy of “per old”variables
along the reference path and the last equality is due to the absence of government
debt along that path. So, like period 1, also the subsequent periods experience a
reduction in national wealth as a consequence of the temporary tax cut in period
0.
Period 1 is special, though. While there is a per capita tax increase by m like
in the subsequent periods, period 1’s old generation still bene…ts from the higher
disposable income in period 0. Hence, in period 2 national wealth per old is even
lower than in period 1 but remains constant henceforth.
A closed economy Also in a closed economy would the future generations be
worse o¤ as a result of a temporary tax cut. Indeed, national wealth (which in
the closed economy equals K) would in period 1 be smaller than otherwise, in
view of the reduced national saving in period 0. And as of period 2 national
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
246
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
wealth would be even smaller than in period 1, in view of the further reduction
in national saving that occurs in period 1.
Summary and perspectives
The fundamental point underlined by OLG models is that there is a di¤erence
between the public sector’s future tax base, including the resources of individuals
yet to be born, and the future tax base emanating from individuals alive today.
This may be called the composition-of-tax-base argument for a tendency to nonneutrality of shifting the timing of (lump-sum) taxation.20
The conclusion that under full capacity utilization budget de…cits imply a
burden for future generations may be seen in a somewhat di¤erent light if persistent technological progress is included in the model. In that case, everything else
equal, future generations will generally be better o¤ than current generations.
Then it might seem less unfair if the former carry some public debt forward to
the latter. In particular this is so if a part of Gt represents government investment
in infrastructure, education, research, health, and environmental protection. As
future generations directly bene…t from such investment, it seems fair that they
also contribute to the …nancing. This is the “bene…ts received principle” known
from public …nance theory.
The above analysis is based on the standard assumption in long-run analysis
that production occurs under full capacity utilization. What if the economy in
period 0 is in an economic depression with high unemployment due to insu¢ cient
aggregate demand? Also in this situation some economists maintain that a cut
in (lump-sum) taxes to stimulate aggregate demand is futile because it has no
real e¤ect. The argument is again that foreseeing the higher taxes needed in
the future, people will save more to prepare themselves (or their descendants
through higher bequests) for paying the higher taxes in the future. The opposite
view is, …rst, that the composition-of-tax-base argument speaks against this as
usual. Second, there is in a depression an additional and quantitatively important
factor. The “…rst-round” increase in consumption due to the temporary tax cut
raises aggregate demand. Thereby production and income is stimulated and a
further (but smaller) rise in consumption occurs in the “second round”and so on
(the Keynesian multiplier process).
This Keynesian mechanism is important for the debate because there are limits
to how large quantitative deviations from Ricardian equivalence the compositionof-tax-base argument alone can deliver. Indeed, taking into account the sizeable
life expectancy of the average citizen, Poterba and Summers (1987) point out that
20
In Exercise 6.?? the reader is asked how the burden of the public debt is distributed across
generations if the debt should be completely wiped out through a tax increase in only periods
1 and 2.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.
6.8. Literature notes
247
the composition-of-tax-base argument by itself delivers only modest deviations if
the issue is timing of taxes over the business cycle.
Another aspect is that in the real world, taxes tend to be distortionary and not
lump sum. On the one hand, this should not be seen as an argument against the
possible theoretical validity of the Ricardian equivalence proposition. The reason
is that Ricardian equivalence (in its strict meaning) claims absence of allocational
e¤ects of changes in the timing of lump-sum taxes. On the other hand, in a wider
perspective the interesting question is, of course, how changes in the timing of
distortionary taxes is likely to a¤ect resource allocation.
Consider …rst income taxes. When taxes are proportional to income or progressive (average tax rate rising in income), they provide insurance through reducing the volatility of after-tax income. The fall in taxes in a recession thus helps
stimulating consumption through reduced precautionary saving (the phenomenon
that current saving tends to rise in response to increased uncertainty, cf. Chapter
30). In this way, replacing lump-sum taxation by income taxation underpins the
positive wealth e¤ect on consumption, arising from the composition-of-tax-base
channel, of a debt-…nanced tax-cut in an economic recession.
What about consumption taxes? A debt-…nanced temporary cut in consumption taxes stimulates consumption through a positive wealth e¤ect, arising from
the composition-of-tax-base channel. On top of this comes a positive intertemporal substitution e¤ect on current consumption caused by the changed consumer
price time pro…le.
The question whether Ricardian non-equivalence is important from a quantitative and empirical point of view pops up in many contexts within macroeconomics. We shall therefore return to the issue several times later in this book.
6.8
Literature notes
(incomplete)
A reader wanting to go more into detail with the acedemic debate about the
design of the EMU and the Stability and Growth Pact is referred to the discussions
in for example Buiter (2003), Buiter and Grafe (2004), Fogel and Saxena (2004),
and Wyplosz (2005). As to discussions of the actual functioning of monetary and
…scal policy in the eurozone in response to the Great Recession see for instance
the opposing views by De Grauwe ( ) and ??.
In addition to the hampering of Keynesian stabilization policy discussed in
Section 6.4.2, also demographic staggering (due to baby booms succeeded by
baby busts) may make rigid de…cit rules problematic. In Denmark for instance
demographic staggering is prognosticated to generate considerable budget de…cits
during several decades after 2030 where younger and smaller generations will succ Groth, Lecture notes in macroeconomics, (mimeo) 2015.
248
CHAPTER 6. LONG-RUN ASPECTS OF FISCAL POLICY
AND PUBLIC DEBT
ceed older and larger ones in the labor market. This is prognosticated to take
place, however, without challenging the long-run sustainability of current …scal
policy as assessed by the Danish Economic Council (see English Summary in De
Økonomiske Råd, 2014). This phenomenon is in Danish known as “hængekøjeproblemet”(the “hammock problem”).
Sources for last part of Section 6.7 ....
6.9
Exercises
6.xx In the OLG model of Section 6.6.2, derive (6.33) and (6.34).
6.? In the OLG model of Section 6.6.2, show that for t = 1; 2; 3; : : : ; public debt
along the “alternative path” evolves according to Bt0 = [(2 + n)=(1 + n)] L1 (1 +
n)t 2 x; where x is the temporary per capita tax cut in period 0. Hint: given
the information in Section 6.6.2 you may start by deriving a …rst-order di¤erence
equation in bt
Bt =Yt with constant coe¢ cients. The information that the
“reference path" has a balanced budget for all t should be taken into account. In
addition, you should explain - and apply - that the initial condition is b1 = B1 =Y1
= (2 + n)x= [f (k)(1 + n)2 ] :
6.?? Consider the OLG model of Section 6.6.2. a) Show that if the temporary
per capita tax cut, x; is su¢ ciently small, the debt can be completely wiped out
through a per capita tax increase in only periods 1 and 2. b) Investigate how in
this case the burden of the debt is distributed across generations. Compare with
the alternative debt policy described in the text.
c Groth, Lecture notes in macroeconomics, (mimeo) 2015.